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@ -102,34 +102,33 @@ We hope that the present review will provide a useful summary of our effort so f
\section{Introduction} \section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Nowadays, there exists a very large number of electronic structure computational approaches, more or less expensive depending on their overall accuracy, able to quantitatively predict the Nowadays, there exist a very large number of electronic structure computational approaches, more or less expensive depending on their overall accuracy, able to quantitatively predict the
absolute and/or relative energies of electronic states in molecular systems \cite{SzaboBook,JensenBook,CramerBook,HelgakerBook}. One important aspect of some of these theoretical absolute and/or relative energies of electronic states in molecular systems \cite{SzaboBook,JensenBook,CramerBook,HelgakerBook}. One important aspect of some of these theoretical
methods is their ability to access the energies of electronic excited states, i.e., states that have higher total energies than the so-called ground (that is, lowest-energy) state methods is their ability to access the energies of electronic excited states, i.e., states that have higher total energies than the so-called ground (that is, lowest-energy) state
\cite{Roos_1996,Piecuch_2002,Dreuw_2005,Krylov_2006,Sneskov_2012,Gonzales_2012,Laurent_2013,Adamo_2013,Ghosh_2018,Blase_2020,Loos_2020a}. \cite{Roos_1996,Piecuch_2002,Dreuw_2005,Krylov_2006,Sneskov_2012,Gonzales_2012,Laurent_2013,Adamo_2013,Ghosh_2018,Blase_2020,Loos_2020a}.
The faithful description of excited states is particularly challenging from a theoretical point of view but is key to a deeper understanding of photochemical and photophysical processes The faithful description of excited states is particularly challenging from a theoretical point of view but is key to a deeper understanding of photochemical and photophysical processes
like absorption, fluorescence, phosphorescence, chemoluminescence, and others \cite{Bernardi_1996,Olivucci_2010,Robb_2007,Navizet_2011,Crespo_2018,Robb_2018,Mai_2020}. like absorption, fluorescence, phosphorescence, chemoluminescence, and others \cite{Bernardi_1996,Olivucci_2010,Robb_2007,Navizet_2011,Crespo_2018,Robb_2018,Mai_2020}.
For a given level of theory, ground-state methods are usually more accurate than their excited-state analogs. For a given level of theory, ground-state methods are usually more accurate than their excited-state analogs.
The reasons behind this are (at least) threefold: i) one might lack a proper variational principle for excited-state energies and one may have to rely on response theory The reasons behind this are (at least) threefold:
\cite{Monkhorst_1977,Helgaker_1989,Koch_1990,Koch_1990b,Christiansen_1995b,Christiansen_1998b,Hattig_2003,Kallay_2004,Hattig_2005c} formalisms which inherently introduce a i) accurately modeling the electronic structure of excited states usually requires larger one-electron basis sets (including diffuse functions most of the times) than their ground-state counterpart,
ground-state ``bias'', ii) accurately modeling the electronic structure of excited states usually requires larger one-electron basis sets (including diffuse functions most of the times) than their ii) excited states can be governed by different amounts of dynamic/static correlations, present very different physical natures ($\pi \to \pis$, $n \to \pis$, charge transfer, double excitation, valence, Rydberg, singlet, doublet, triplet, etc), yet be very close in energy from one another, and
ground-state counterpart, and iii) excited states can be governed by different amounts of dynamic/static correlations, present very different physical natures ($\pi \to \pis$, $n \to \pis$, charge iii) one usually has to rely on response theory formalisms \cite{Monkhorst_1977,Helgaker_1989,Koch_1990,Koch_1990b,Christiansen_1995b,Christiansen_1998b,Hattig_2003,Kallay_2004,Hattig_2005c}, which inherently introduce a ground-state ``bias''.
transfer, double excitation, valence, Rydberg, singlet, doublet, triplet, etc), yet be very close in energy from one another. Hence, designing excited-state methods able to tackle simultaneously Hence, designing excited-state methods able to tackle simultaneously and on an equal footing all these types of excited states at an affordable cost remains an open challenge in theoretical computational chemistry as evidenced by the large number of review
and on an equal footing all these types of excited states at an affordable cost remains an open challenge in theoretical computational chemistry as evidenced by the large number of review
articles on this particular subject \cite{Roos_1996,Piecuch_2002,Dreuw_2005,Krylov_2006,Sneskov_2012,Gonzales_2012,Laurent_2013,Adamo_2013,Dreuw_2015,Ghosh_2018,Blase_2020,Loos_2020a}. articles on this particular subject \cite{Roos_1996,Piecuch_2002,Dreuw_2005,Krylov_2006,Sneskov_2012,Gonzales_2012,Laurent_2013,Adamo_2013,Dreuw_2015,Ghosh_2018,Blase_2020,Loos_2020a}.
When designing a new theoretical model, the first feature that one might want to test is its overall accuracy, i.e., its ability to reproduce reference (or benchmark) values for a given system with a well-defined When designing a new theoretical model, the first feature that one might want to test is its overall accuracy, i.e., its ability to reproduce reference (or benchmark) values for a given system with a well-defined
setup (same geometry, basis set, etc). These values can be absolute and/or relative energies, geometrical parameters, physical or chemical spectroscopic properties extracted from experiments, setup (same geometry, basis set, etc). These values can be absolute and/or relative energies, geometrical parameters, physical or chemical spectroscopic properties extracted from experiments,
high-level theoretical calculations, or any combination of these. To this end, the electronic structure community has designed along the years benchmark sets, i.e., sets of molecules for which one high-level theoretical calculations, or any combination of these. To this end, the electronic structure community has designed along the years benchmark sets, i.e., sets of molecules for which one
could (very) accurately compute theoretical estimates and/or access solid experimental data for given properties. Regarding ground-states properties, two of the oldest and most employed sets are can (very) accurately compute theoretical estimates and/or access solid experimental data for given properties. Regarding ground-states properties, two of the oldest and most employed sets are
probably the Gaussian-1 and Gaussian-2 benchmark sets \cite{Pople_1989,Curtiss_1991,Curtiss_1997} developed by the group of Pople in the 1990's. For example, the Gaussian-2 set gathers atomization probably the Gaussian-1 and Gaussian-2 benchmark sets \cite{Pople_1989,Curtiss_1991,Curtiss_1997} developed by the group of Pople in the 1990's. For example, the Gaussian-2 set gathers atomization
energies, ionization energies, electron affinities, proton affinities, bond dissociation energies, and reaction barriers. This set was subsequently extended and refined \cite{Curtiss_1998,Curtiss_2007}. energies, ionization energies, electron affinities, proton affinities, bond dissociation energies, and reaction barriers. This set was subsequently extended and refined \cite{Curtiss_1998,Curtiss_2007}.
Another very useful set for the design of methods able to catch dispersion effects \cite{Angyan_2020} is the S22 benchmark set \cite{Jureka_2006} (and its extended S66 version \cite{Rezac_2011}) Another very useful set for the design of methods able to catch dispersion effects \cite{Angyan_2020} is the S22 benchmark set \cite{Jureka_2006} (and its extended S66 version \cite{Rezac_2011})
of Hobza and collaborators which provides benchmark interaction energies for weakly-interacting (non covalent) systems. One could also mentioned the $GW$100 set \cite{vanSetten_2015,Krause_2015,Maggio_2016} of Hobza and collaborators which provides benchmark interaction energies for weakly-interacting (non covalent) systems. One could also mentioned the $GW$100 set \cite{vanSetten_2015,Krause_2015,Maggio_2016}
(and its $GW$5000 extension \cite{Stuke_2020}) of ionization energies which has helped enormously the community to compare the implementation of $GW$-type methods for molecular (and its $GW$5000 extension \cite{Stuke_2020}) of ionization energies which has helped enormously the community to compare the implementation of $GW$-type methods for molecular
systems \cite{vanSetten_2013,Bruneval_2016,Caruso_2016,Govoni_2018}. The extrapolated ab initio thermochemistry (HEAT) set designed to achieve high accuracy for enthalpies of formation systems \cite{vanSetten_2013,Bruneval_2016,Caruso_2016,Govoni_2018}. The extrapolated ab initio thermochemistry (HEAT) set designed to achieve high accuracy for enthalpies of formation
of atoms and small molecules (without experimental data) is yet another successful example of benchmark set \cite{Tajti_2004,Bomble_2006,Harding_2008}. More recently, the benchmark datasets of atoms and small molecules (without experimental data) is yet another successful example of benchmark set \cite{Tajti_2004,Bomble_2006,Harding_2008}. More recently, let us mention the benchmark datasets
provided by the \textit{Simons Collaboration on the Many-Electron Problem} have been extremely valuable to the community by providing, for example, highly-accurate ground-state energies for of the \textit{Simons Collaboration on the Many-Electron Problem} providing, for example, highly-accurate ground-state energies for
hydrogen chains \cite{Motta_2017} as well as transition metal atoms and their ions and monoxides \cite{Williams_2020}. Let us also mention the set of Zhao and Truhlar for small transition metal complexes hydrogen chains \cite{Motta_2017} as well as transition metal atoms and their ions and monoxides \cite{Williams_2020}. Let us also mention the set of Zhao and Truhlar for small transition metal complexes
employed to compare the accuracy of density-functional methods \cite{ParrBook} for $3d$ transition-metal chemistry \cite{Zhao_2006}, and finally the popular GMTKN24 \cite{Goerigk_2010}, employed to compare the accuracy of density-functional methods \cite{ParrBook} for $3d$ transition-metal chemistry \cite{Zhao_2006}, and finally the popular GMTKN24 \cite{Goerigk_2010},
GMTKN30 \cite{Goerigk_2011a,Goerigk_2011b} and GMTKN55 \cite{Goerigk_2017} databases for general main group thermochemistry, kinetics, and non-covalent interactions developed by Goerigk, Grimme and GMTKN30 \cite{Goerigk_2011a,Goerigk_2011b} and GMTKN55 \cite{Goerigk_2017} databases for general main group thermochemistry, kinetics, and non-covalent interactions developed by Goerigk, Grimme and
@ -152,7 +151,7 @@ as well as the Gagliardi-Truhlar set employed to compare the accuracy of multico
Following a similar philosophy and striving for chemical accuracy, we have recently reported in several studies highly-accurate vertical excitations for small- and medium-sized molecules Following a similar philosophy and striving for chemical accuracy, we have recently reported in several studies highly-accurate vertical excitations for small- and medium-sized molecules
\cite{Loos_2020a,Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c}. The so-called QUEST dataset of vertical excitations which we will describe in detail in the present review article is composed by 5 \cite{Loos_2020a,Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c}. The so-called QUEST dataset of vertical excitations which we will describe in detail in the present review article is composed by 5
subsets (see Fig.~\ref{fig:scheme}): i) a subset of excitations in small molecules containing from 1 to 3 non-hydrogen atoms known as QUEST\#1, ii) a subset of double excitations for molecules of small and subsets (see Fig.~\ref{fig:scheme}): i) a subset of excitations in small molecules containing from 1 to 3 non-hydrogen atoms known as QUEST\#1, ii) a subset of double excitations in molecules of small and
medium sizes known as QUEST\#2, iii) a subset of excitation energies for medium-sized molecules containing from 4 to 6 non-hydrogen atoms known as QUEST\#3, iv) a subset composed by more ``exotic'' medium sizes known as QUEST\#2, iii) a subset of excitation energies for medium-sized molecules containing from 4 to 6 non-hydrogen atoms known as QUEST\#3, iv) a subset composed by more ``exotic''
molecules and radicals labeled as QUEST\#4, and v) a subset known as QUEST\#5, specifically designed for the present article, gathering excitation energies in larger molecules as well as additional smaller molecules. molecules and radicals labeled as QUEST\#4, and v) a subset known as QUEST\#5, specifically designed for the present article, gathering excitation energies in larger molecules as well as additional smaller molecules.
One of the key aspect of the QUEST dataset is that it does not rely on any experimental values, avoiding potential biases inherently linked to experiments and facilitating in the process theoretical comparisons. One of the key aspect of the QUEST dataset is that it does not rely on any experimental values, avoiding potential biases inherently linked to experiments and facilitating in the process theoretical comparisons.
@ -171,7 +170,7 @@ review of the generic benchmark studies devoted to adiabatic and 0-0 energies pe
The QUEST dataset has the particularity to be based to a large extent on selected configuration interaction (SCI) reference excitation energies as well as high-order linear-response (LR) CC methods such as LR-CCSDT and The QUEST dataset has the particularity to be based to a large extent on selected configuration interaction (SCI) reference excitation energies as well as high-order linear-response (LR) CC methods such as LR-CCSDT and
LR-CCSDTQ \cite{Noga_1987,Koch_1990,Kucharski_1991,Christiansen_1998b,Kucharski_2001,Kowalski_2001,Kallay_2003,Kallay_2004,Hirata_2000,Hirata_2004}. Recently, SCI methods have been a force to reckon with for LR-CCSDTQ \cite{Noga_1987,Koch_1990,Kucharski_1991,Christiansen_1998b,Kucharski_2001,Kowalski_2001,Kallay_2003,Kallay_2004,Hirata_2000,Hirata_2004}. Recently, SCI methods have been a force to reckon with for
the computation of highly-accurate energies in small- and medium-sized molecules as they yield near full configuration interaction (FCI) quality energies for only a fraction of the computational cost of a genuine FCI calculation \cite{Booth_2009,Booth_2010,Cleland_2010,Booth_2011,Daday_2012,Blunt_2015,Ghanem_2019,Deustua_2017,Deustua_2018,Holmes_2017,Chien_2018,Li_2018,Yao_2020,Li_2020,Eriksen_2017,Eriksen_2018,Eriksen_2019a,Eriksen_2019b,Xu_2018,Xu_2020,Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c,Loos_2020a,Loos_2020e,Eriksen_2021}. the computation of highly-accurate energies in small- and medium-sized molecules as they yield near full configuration interaction (FCI) quality energies for only a very tiny fraction of the computational cost of a genuine FCI calculation \cite{Booth_2009,Booth_2010,Cleland_2010,Booth_2011,Daday_2012,Blunt_2015,Ghanem_2019,Deustua_2017,Deustua_2018,Holmes_2017,Chien_2018,Li_2018,Yao_2020,Li_2020,Eriksen_2017,Eriksen_2018,Eriksen_2019a,Eriksen_2019b,Xu_2018,Xu_2020,Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c,Loos_2020a,Loos_2020e,Eriksen_2021}.
Due to the fairly natural idea underlying these methods, the SCI family is composed of numerous members \cite{Bender_1969,Whitten_1969,Huron_1973,Abrams_2005,Bunge_2006,Bytautas_2009,Giner_2013,Caffarel_2014,Giner_2015,Garniron_2017b,Caffarel_2016a,Caffarel_2016b,Holmes_2016,Sharma_2017,Holmes_2017,Chien_2018,Scemama_2018,Scemama_2018b,Garniron_2018,Evangelista_2014,Tubman_2016,Tubman_2020,Schriber_2016,Schriber_2017,Liu_2016,Per_2017,Ohtsuka_2017,Zimmerman_2017,Li_2018,Ohtsuka_2017,Coe_2018,Loos_2019}. Due to the fairly natural idea underlying these methods, the SCI family is composed of numerous members \cite{Bender_1969,Whitten_1969,Huron_1973,Abrams_2005,Bunge_2006,Bytautas_2009,Giner_2013,Caffarel_2014,Giner_2015,Garniron_2017b,Caffarel_2016a,Caffarel_2016b,Holmes_2016,Sharma_2017,Holmes_2017,Chien_2018,Scemama_2018,Scemama_2018b,Garniron_2018,Evangelista_2014,Tubman_2016,Tubman_2020,Schriber_2016,Schriber_2017,Liu_2016,Per_2017,Ohtsuka_2017,Zimmerman_2017,Li_2018,Ohtsuka_2017,Coe_2018,Loos_2019}.
Their fundamental philosophy consists, roughly speaking, in retaining only the most relevant determinants of the FCI space following a given criterion to slow down the exponential increase of the size of the CI expansion. Their fundamental philosophy consists, roughly speaking, in retaining only the most relevant determinants of the FCI space following a given criterion to slow down the exponential increase of the size of the CI expansion.
Originally developed in the late 1960's by Bender and Davidson \cite{Bender_1969} as well as Whitten and Hackmeyer \cite{Whitten_1969}, new efficient SCI algorithms have resurfaced recently. Originally developed in the late 1960's by Bender and Davidson \cite{Bender_1969} as well as Whitten and Hackmeyer \cite{Whitten_1969}, new efficient SCI algorithms have resurfaced recently.
@ -233,7 +232,7 @@ For open-shell molecules, the CCSDT, CCSDTQ, and CCSDTQP calculations performed
All excited-state calculations are performed, except when explicitly mentioned, in the frozen-core (FC) approximation using large cores for the third-row atoms. All excited-state calculations are performed, except when explicitly mentioned, in the frozen-core (FC) approximation using large cores for the third-row atoms.
All the SCI calculations are performed within the frozen-core approximation using QUANTUM PACKAGE \cite{Garniron_2019} where the CIPSI algorithm \cite{Huron_1973} is implemented. Details regarding this specific CIPSI implementation can be found in Refs.~\cite{Garniron_2019} and \cite{Scemama_2019}. All the SCI calculations are performed within the frozen-core approximation using QUANTUM PACKAGE \cite{Garniron_2019} where the CIPSI algorithm \cite{Huron_1973} is implemented. Details regarding this specific CIPSI implementation can be found in Refs.~\cite{Garniron_2019} and \cite{Scemama_2019}.
A state-averaged formalism is employed, i.e., the ground and excited states are described with the same set of determinants, but different CI coefficients. A state-averaged formalism is employed, i.e., the ground and excited states are described with the same set of determinants and orbitals, but different CI coefficients.
Our usual protocol \cite{Scemama_2018,Scemama_2018b,Scemama_2019,Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c} consists of performing a preliminary CIPSI calculation using Hartree-Fock orbitals in order to generate a CIPSI wave function with at least $10^7$ determinants. Our usual protocol \cite{Scemama_2018,Scemama_2018b,Scemama_2019,Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c} consists of performing a preliminary CIPSI calculation using Hartree-Fock orbitals in order to generate a CIPSI wave function with at least $10^7$ determinants.
Natural orbitals are then computed based on this wave function, and a new, larger CIPSI calculation is performed with this new set of orbitals. Natural orbitals are then computed based on this wave function, and a new, larger CIPSI calculation is performed with this new set of orbitals.
This has the advantage to produce a smoother and faster convergence of the SCI energy toward the FCI limit. This has the advantage to produce a smoother and faster convergence of the SCI energy toward the FCI limit.
@ -394,7 +393,7 @@ Tetrazine & $^1B_{3u}(n \ra \pis)$ & 2.53 & 2.54 & 2.56(5) & 5.07(16) \\
Pyridazine & $^1B_1(n \ra \pis)$ & 3.95 & 3.95 & 3.97(10)& 3.60(43) \\ Pyridazine & $^1B_1(n \ra \pis)$ & 3.95 & 3.95 & 3.97(10)& 3.60(43) \\
& $^3B_1(n \ra \pis)$ & 3.27 & 3.26 & 3.27(15)& 3.46(14) \\ & $^3B_1(n \ra \pis)$ & 3.27 & 3.26 & 3.27(15)& 3.46(14) \\
Pyridine & $^1B_1(n \ra \pis)$ & 5.12 & 5.10 & 5.15(12)& 4.90(24) \\ Pyridine & $^1B_1(n \ra \pis)$ & 5.12 & 5.10 & 5.15(12)& 4.90(24) \\
& $^3A_1(\pi \ra \pis)$ & 4.33 & 4.31 & 4.42(85)& 3.68(1.05) \\ & $^3A_1(\pi \ra \pis)$ & 4.33 & 4.31 & 4.42(85)& 3.68(105) \\
Pyrimidine & $^1B_1(n \ra \pis)$ & 4.58 & 4.57 & 4.64(11)& 2.54(5) \\ Pyrimidine & $^1B_1(n \ra \pis)$ & 4.58 & 4.57 & 4.64(11)& 2.54(5) \\
& $^3B_1(n \ra \pis)$ & 4.20 & 4.20 & 4.55(37)& 2.18(27) \\ & $^3B_1(n \ra \pis)$ & 4.20 & 4.20 & 4.55(37)& 2.18(27) \\
Triazine & $^1A_1''(n \ra \pis)$ & 4.85 & 4.84 & 4.77(13)& 5.12(51) \\ Triazine & $^1A_1''(n \ra \pis)$ & 4.85 & 4.84 & 4.77(13)& 5.12(51) \\
@ -403,7 +402,7 @@ Triazine & $^1A_1''(n \ra \pis)$ & 4.85 & 4.84 & 4.77(13)& 5.12(51) \\
\hline % Please only put a hline at the end of the table \hline % Please only put a hline at the end of the table
\end{tabular} \end{tabular}
\begin{tablenotes} \begin{tablenotes}
\item $^a$ Excitation energies and error bars estimated via the present method based on Gaussian random variables (see Sec.~\ref{sec:error}). \item $^a$ Excitation energies and error bars estimated via the novel statistical method based on Gaussian random variables (see Sec.~\ref{sec:error}).
The error bars reported in parenthesis correspond to one standard deviation. The error bars reported in parenthesis correspond to one standard deviation.
\item $^b$ Excitation energies obtained via a three-point linear fit using the three largest CIPSI variational wave functions, and error bars estimated via the extrapolation distance, \ie, the difference in excitation energies obtained with the three-point linear extrapolation and the largest CIPSI wave function. \item $^b$ Excitation energies obtained via a three-point linear fit using the three largest CIPSI variational wave functions, and error bars estimated via the extrapolation distance, \ie, the difference in excitation energies obtained with the three-point linear extrapolation and the largest CIPSI wave function.
\end{tablenotes} \end{tablenotes}
@ -1245,8 +1244,8 @@ via GitHub pull requests.
In the present review article, we have presented and extended the QUEST database of highly-accurate excitation energies for molecular systems \cite{Loos_2020a,Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c} that we started building In the present review article, we have presented and extended the QUEST database of highly-accurate excitation energies for molecular systems \cite{Loos_2020a,Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c} that we started building
in 2018 and that is now composed by more than 500 vertical excitations, many of which can be reasonably considered as within 1 kcal/mol (or less) of the FCI limit for the considered CC3/aug-cc-pVTZ geometry and basis set (\emph{aug}-cc-pVTZ). in 2018 and that is now composed by more than 500 vertical excitations, many of which can be reasonably considered as within 1 kcal/mol (or less) of the FCI limit for the considered CC3/aug-cc-pVTZ geometry and basis set (\emph{aug}-cc-pVTZ).
In particular, we have detailed the specificities of our protocol by providing computational details regarding geometries, basis sets, as well as reference and benchmarked computational methods. The content of our five QUEST subsets has In particular, we have detailed the specificities of our protocol by providing computational details regarding geometries, basis sets, as well as reference and benchmarked computational methods. The content of our five QUEST subsets has
been presented in detail, and for each of them, we have provided the number of reference excitation energies, the nature and size of the molecules, the list of benchmarked methods, as well as other useful specificities. Importantly, we have been presented in detail, and for each of them, we have provided the number of reference excitation energies, the nature and size of the molecules, the list of benchmarked methods, as well as other useful specificities.
proposed a new method to faithfully estimate the extrapolation error in SCI calculations. This new method based on Gaussian random variables has been tested by computing additional FCI values for five- and six-membered rings. Importantly, we have proposed a new statistical method that produces much safer estimates of the extrapolation error in SCI calculations. This new method based on Gaussian random variables has been tested by computing additional FCI values for five- and six-membered rings.
After having discussed the generation of our TBEs, we have reported a comprehensive benchmark for a significant number of methods on the entire QUEST set with, in addition, a specific analysis for each type of excited states. After having discussed the generation of our TBEs, we have reported a comprehensive benchmark for a significant number of methods on the entire QUEST set with, in addition, a specific analysis for each type of excited states.
Finally, the main features of the website specifically designed to gather the entire data generated during these past few years have been presented and discussed. Finally, the main features of the website specifically designed to gather the entire data generated during these past few years have been presented and discussed.
@ -1256,7 +1255,7 @@ In this framework, we provide in the {\SupInf} a file with all our benchmark dat
Regarding future improvements and extensions, we would like to mention that although our present goal is to produce chemically accurate vertical excitation energies, we are currently devoting great efforts to obtain highly-accurate excited-state properties \cite{Hodecker_2019,Eriksen_2020b} such as dipoles and oscillator strengths for molecules of small and medium sizes \cite{Chrayteh_2021,Sarkar_2021}, so as to complete previous efforts aiming at determining accurate excited-state geometries \cite{Budzak_2017,Jacquemin_2018}. Regarding future improvements and extensions, we would like to mention that although our present goal is to produce chemically accurate vertical excitation energies, we are currently devoting great efforts to obtain highly-accurate excited-state properties \cite{Hodecker_2019,Eriksen_2020b} such as dipoles and oscillator strengths for molecules of small and medium sizes \cite{Chrayteh_2021,Sarkar_2021}, so as to complete previous efforts aiming at determining accurate excited-state geometries \cite{Budzak_2017,Jacquemin_2018}.
Reference ground-state properties (such as correlation energies and atomization energies) are also being currently produced \cite{Scemama_2020,Loos_2020e}. Reference ground-state properties (such as correlation energies and atomization energies) are also being currently produced \cite{Scemama_2020,Loos_2020e}.
Besides this, because computing 500 (or so) excitation energies can be a costly exercise even with cheap computational methods, we are planning on developing a ``diet set'' following the philosophy of the ``diet GMTKN55'' set proposed recently by Gould \cite{Gould_2018b}. Besides this, because computing 500 (or so) excitation energies can be a costly exercise even with cheap computational methods, we are planning on developing a ``diet set'' (\ie, a much smaller set of excitation energies which can reproduce key results of the full QUEST database, including ranking of approximations) following the philosophy of the ``diet GMTKN55'' set proposed recently by Gould \cite{Gould_2018b}.
We hope to report on this in the near future. We hope to report on this in the near future.
%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -1267,7 +1266,6 @@ AS, MC, and PFL thank the European Research Council (ERC) under the European Uni
Funding from the \textit{``Centre National de la Recherche Scientifique''} is also acknowledged. Funding from the \textit{``Centre National de la Recherche Scientifique''} is also acknowledged.
DJ acknowledges the \textit{R\'egion des Pays de la Loire} for financial support and the CCIPL computational center for ultra-generous allocation of computational time. DJ acknowledges the \textit{R\'egion des Pays de la Loire} for financial support and the CCIPL computational center for ultra-generous allocation of computational time.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{conflict of interest} \section*{conflict of interest}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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Manuscript/SI/QUEST_WIREs-SI.tex
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@ -110,7 +110,7 @@ and the $^1B_{3u}$ states) being considered as ``unsafe'' in the database.
Comparisons to experimental 0-0 energies in condensed medium and CNDO calculations can be found in the Table, but are not very helpful to assess our TBEs. Comparisons to experimental 0-0 energies in condensed medium and CNDO calculations can be found in the Table, but are not very helpful to assess our TBEs.
To the best of our knowledge, the present work is the first to report triplet excited states, and we list in Table \ref{tab:azanaph} eight valence transitions obtained at the CC3/{\AVTZ} level. As we were not able to To the best of our knowledge, the present work is the first to report triplet excited states, and we list in Table \ref{tab:azanaph} eight valence transitions obtained at the CC3/{\AVTZ} level. As we were not able to
perform CCSDT calculations for the triplets, all these transition energies are labeled ``unsafe'' in the QUEST database. perform CCSDT calculations for the triplets, all these transition energies are labeled ``unsafe'' in the QUEST database.
Nevertheless, given the large \%$T_1$ values, one can likely consider them accurate (for a given basis set at least). Nevertheless, given the large \%$T_1$ values, one can likely consider them accurate (for the basis set used at least).
% %
%%% TABLE %%% %%% TABLE %%%
@ -365,7 +365,7 @@ The $A_g$ transitions are known to be much more challenging: the states are dark
On a positive note, the basis set effects are very limited for the $A_g$ state, {\Pop} being apparently sufficient. On a positive note, the basis set effects are very limited for the $A_g$ state, {\Pop} being apparently sufficient.
In contrast, as expected for such transition, there is a significant drop of the theoretical estimate in going from CC3 to CCSDT. In contrast, as expected for such transition, there is a significant drop of the theoretical estimate in going from CC3 to CCSDT.
From the analysis performed for double excitations in Ref.~\cite{Loos_2019}, it is unclear if NEVPT2 or CASPT2 would in fact outperform CCSDT for such ``mixed-character'' state, so that we cannot define a trustworthy TBE on this basis. From the analysis performed for double excitations in Ref.~\cite{Loos_2019}, it is unclear if NEVPT2 or CASPT2 would in fact outperform CCSDT for such ``mixed-character'' state, so that we cannot define a trustworthy TBE on this basis.
However, based on our experience for butadiene \cite{Loos_2020b}, one can widely estimate the transition energies to be in the range 5.55--5.60 eV for hexatriene and in the range 4.80--4.85 eV for octratetraene. However, based on our experience for butadiene \cite{Loos_2020b}, one can widely estimate the transition energy to be in the range 5.55--5.60 eV for hexatriene and in the range 4.80--4.85 eV for octratetraene.
Interestingly the FCI value of Chien \textit{et al.}~with a small basis set for hexatriene (5.59 eV) is compatible with such an estimate. Interestingly the FCI value of Chien \textit{et al.}~with a small basis set for hexatriene (5.59 eV) is compatible with such an estimate.
Experimentally, for hexatriene, multiphoton experiments estimate the $A_g$ state to be slightly above the $B_u$ transition \cite{Fujii_1985}, an outcome that theory reproduces. Experimentally, for hexatriene, multiphoton experiments estimate the $A_g$ state to be slightly above the $B_u$ transition \cite{Fujii_1985}, an outcome that theory reproduces.

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